Dynamical Systems in Neuroscience - Table of Contents

November 2006
7 x 10, 457 pp., 409 illus.
$62.00/£45.95 (CLOTH)
Short

ISBN-10:
0-262-09043-0
ISBN-13:
978-0-262-09043-8

Paper (2010)

Computational Neuroscience

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Dynamical Systems in Neuroscience
The Geometry of Excitability and Bursting
Eugene M. Izhikevich

	Preface	xv

1.	Introduction Sample Chapter - Download PDF (381 KB)	1

1.1	Neurons	1

1.1.1	What Is a Spike?	2

1.1.2	Where Is the Threshold?	3

1.1.3	Why Are Neurons Different, and Why Do We Care?	6

1.1.4	Building Models	6

1.2	Dynamical Systems	8

1.2.1	Phase Portraits	8

1.2.2	Bifurcations	11

1.2.3	Hodgkin Classification	14

1.2.4	Neurocomputational properties	16

1.2.5	Building Models (Revisited)	20

	Review of Important Concepts	21

	Bibliographical Notes	21

2.	Electrophysiology of Neurons	25

2.1	Ions	25

2.1.1	Nernst Potential	26

2.1.2	Ionic Currents and Conductances	27

2.1.3	Equivalent Circuit	28

2.1.4	Resting Potential and Input Resistance	29

2.1.5	Voltage-Clamp and I-V Relation	30

2.2	Conductances	32

2.2.1	Voltage-Gated Channels	33

2.2.2	Activation of Persistent Currents	34

2.2.3	Inactivation of Transient Currents	35

2.2.4	Hyperpolarization-Activated Channels	36

2.3	The Hodgkin-Huxley Model	37

2.3.1	Hodgkin-Huxley Equations	37

2.3.2	Action Potential	41

2.3.3	Propagation of the Action Potentials	42

2.3.4	Dendritic Compartments	43

2.3.5	Summary of Voltage-Gated Currents	44

	Review of Important Concepts	49

	Bibliographical Notes	50

	Exercises	50

3.	One-Dimensional Systems	53

3.1	Electrophysiological Examples	53

3.1.1	I-V Relations and Dynamics	54

3.1.2	Leak + Instantaneous I_{Na, p}	55

3.2	Dynamical Systems	57

3.2.1	Geometrical Analysis	59

3.2.2	Equilibria	60

3.2.3	Stability	60

3.2.4	Eigenvalues	61

3.2.5	Unstable Equilibria	61

3.2.6	Attraction Domain	62

3.2.7	Threshold and Action Potential	63

3.2.7	Threshold and Action Potential	63

3.2.8	Bistability and Hysteresis	66

3.3	Phase Portraits	67

3.3.1	Topological Equivalence	68

3.3.2	Local Equivalence and the Hartman-Grobman Theorem	69

3.3.3	Bifurcations	70

3.3.4	Saddle-Node (Fold) Bifurcation	74

3.3.5	Slow Transition	75

3.3.6	Bifurcation Diagram	77

3.3.7	Bifurcations and I-V Relations	77

3.3.8	Quadratic Integrate-and-Fire Neuron	80

	Review of Important Concepts	82

	Bibliographical Notes	83

	Exercises	83

4.	Two-Dimensional Systems	89

4.1	Planar Vector Fields	89

4.1.1	Nullclines	92

4.1.2	Trajectories	94

4.1.3	Limit Cycles	96

4.1.4	Relaxation Oscillators	98

4.2	Equilibria	99

4.2.1	Stability	100

4.2.2	Local Linear Analysis	101

4.2.3	Eigenvalues and Eigenvectors	102

4.2.4	Local Equivalence	103

4.2.5	Classification of Equilibria	103

4.2.6	Example: FitzHugh-Nagumo Model	106

4.3	Phase Portraits	108

4.3.1	Bistability and Attraction Domains	108

4.3.2	Stable/Unstable Manifolds	109

4.3.3	Homoclinic/Heteroclinic Trajectories	111

4.3.4	Saddle-Node Bifurcation	113

4.3.5	Andronov-Hopf Bifurcation	116

	Review of Important Concepts	121

	Bibliographical Notes	122

	Exercises	122

5.	Conductance-Based Models and Their Reductions	127

5.1	Minimal Models	127

5.1.1	Amplifying and Resonant Gating Variables	129

5.1.2	I_Na,p+I_K -Model	132

5.1.3	I_Na,t -Model	133

5.1.4	I_{Na, p}+I_h -Model	136

5.1.5	I_h+I_Kir -Model	138

5.1.6	I_K+I_Kir -Model	140

5.1.7	I_A -Model	142

5.1.8	Ca²⁺ -Gated Minimal Models	147

5.2	Reduction of Multidimensional Models	147

5.2.1	Hodgkin-Huxley model	147

5.2.2	Equivalent Potentials	151

5.2.3	Nullclines and I-V Relations	151

5.2.4	Reduction to Simple Model	153

	Review of Important Concepts	156

	Bibliographical Notes	156

	Exercises	157

6.	Bifurcations	159

6.1	Equilibrium (Rest State)	159

6.1.1	Saddle-Node (Fold)	162

6.1.2	Saddle-Node on Invariant Circle	164

6.1.3	Supercritical Andronov-Hopf	168

6.1.4	Subcritical Andronov-Hopf	174

6.2	Limit Cycle (Spiking State)	178

6.2.1	Saddle-Node on Invariant Circle	180

6.2.2	Supercritical Andronov-Hopf	181

6.2.3	Fold Limit Cycle	181

6.2.4	Homoclinic	185

6.3	Other Interesting Cases	190

6.3.1	Three-Dimensional Phase Space	190

6.3.2	Cusp and Pitchfork	192

6.3.3	Bogdanov-Takens	194

6.3.4	Relaxation Oscillators and Canards	198

6.3.5	Bautin	200

6.3.6	Saddle-Node Homoclinic Orbit	201

6.3.7	Hard and Soft Loss of Stability	204

	Bibliographical Notes	205

	Exercises	210

7.	Neuronal Excitability	215

7.1	Excitability	215

7.1.1	Bifurcations	216

7.1.2	Hodgkin's Classification	218

7.1.3	Classes 1 and 2	221

7.1.4	Class 3	222

7.1.5	Ramps, Steps, and Shocks	224

7.1.6	Bistability	226

7.1.7	Class 1 and 2 Spiking	228

7.2	Integrators vs. Resonators	229

7.2.1	Fast Subthreshold Oscillations	230

7.2.2	Frequency Preference and Resonance	232

7.2.3	Frequency Preference in Vivo	237

7.2.4	Thresholds and Action Potentials	238

7.2.5	Threshold Manifolds	240

7.2.6	Rheobase	242

7.2.7	Postinhibitory Spike	242

7.2.8	Inhibition-Induced Spiking	244

7.2.9	Spike Latency	246

7.2.10	Flipping from an Integrator to a Resonator	248

7.2.11	Transition Between Integrators and Resonators	251

7.3	Slow Modulation	252

7.3.1	Spike Frequency Modulation	255

7.3.2	I-V Relation	256

7.3.3	Slow Subthreshold Oscillation	258

7.3.4	Rebound Response and Voltage Sag	259

7.3.5	AHP and ADP	260

	Review of Important Concepts	264

	Bibliographical Notes	264

	Exercises	265

8.	Simple Models	267

8.1	Simplest Models	267

8.1.1	Integrate-and-Fire	268

8.1.2	Resonate-and-Fire	269

8.1.3	Quadratic Integrate-and-Fire	270

8.1.4	Simple Model of Choice	272

8.1.5	Canonical Models	278

8.2	Cortex	281

8.2.1	Regular Spiking (RS) Neurons	282

8.2.2	Intrinsically Bursting (IB) Neurons	288

8.2.3	Multi-Compartment Dendritic Tree	292

8.2.4	Chattering (CH) Neurons	294

8.2.5	Low-Threshold Spiking (LTS) Interneurons	296

8.2.6	Fast Spiking (FS) Interneurons	298

8.2.7	Late Spiking (LS) Interneurons	300

8.2.8	Diversity of Inhibitory Interneurons	301

8.3	Thalamus	304

8.3.1	Thalamocortical (TC) Relay Neurons	305

8.3.2	Reticular Thalamic Nucleus (RTN) Neurons	306

8.3.3	Thalamic Interneurons	308

8.4	Other Interesting Cases	308

8.4.1	Hippocampal CA1 Pyramidal Neurons	308

8.4.2	Spiny Projection Neurons of Neostriatum and Basal Ganglia	311

8.4.3	Mesencephalic V Neurons of Brainstream	313

8.4.4	Stellate Cells of Entorhinal Cortex	314

8.4.5	Mital Neurons of the Olfactory Bulb	316

	Review of Important Concepts	319

	Bibliographical Notes	319

	Exercises	321

9.	Bursting	325

9.1	Electrophysiology	325

9.1.1	Example: The I_Na,p+I_K+I_K(M)-Model	327

9.1.2	Fast-Slow Dynamics	329

9.1.3	Minimal Models	332

9.1.4	Central Pattern Generators and Half-Center Oscillators	334

9.2	Geometry	335

9.2.1	Fast-Slow Bursters	336

9.2.2	Phase Portraits	336

9.2.3	Averaging	339

9.2.4	Equivalent Voltage	341

9.2.5	Hysteresis Loops and Slow Waves	342

9.2.6	Bifurcations "Resting <--> Bursting <--> Tonic Spiking"	344

9.3	Classification	347

9.3.1	Fold/Homoclinic	350

9.3.2	Circle/Circle	354

9.3.4	Fold/Fold Cycle	364

9.3.5	Fold/Hopf	365

9.3.6	Fold/Circle	366

9.4	Neurocomputational Properties	367

9.4.1	How to Distinguish?	367

9.4.2	Integrators vs. Resonators	368

9.4.3	Bistability	368

9.4.4	Bursts as a Unit of Neuronal Information	371

9.4.5	Chirps	372

9.4.6	Synchronization	373

	Review of Important Concepts	375

	Bibliographical Notes	376

	Exercises	378

10.	Synchronization	385

	Solutions to Exercises	387

	References	419

	Index	435

	Synchronization (www.izhikevich.com)	443

10.1	Pulsed Coupling	444

10.1.1	Phase of Oscillation	444

10.1.2	Isochrons	445

10.1.3	PRC	446

10.1.4	Type 0 and Type 1 Phase Response	450

10.1.5	Poincare Phase Map	452

10.1.6	Fixed Points	453

10.1.7	Synchronization	454

10.1.8	Phase-Locking	456

10.1.9	Arnold Tongues	456

10.2	Weak Coupling	458

10.2.1	Winfree's Approach	459

10.2.2	Kuramoto's Approach	460

10.2.3	Malkin's Approach	461

10.2.4	Measuring PRCs Experimentally	462

10.2.5	Phase Model for Coupled Oscillators	465

10.3	Synchronization	467

10.3.1	Two Oscillators	469

10.3.2	Chains	471

10.3.3	Networks	473

10.3.4	Mean-Field Approximations	474

10.4	Examples	475

10.4.1	Phase Oscillators	475

10.4.2	SNIC Oscillators	477

10.4.3	Homoclinic Oscillators	482

10.4.4	Relaxation Oscillators and FTM	484

10.4.5	Bursting Oscillators	486

	Review of Important Concepts	488

	Bibliographical Notes	489

	Solutions	497

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